PAMO 2023 Problem 3
Source: 2023 Pan African Mathematics Olympiad
May 17, 2023
algebraPAMO
Problem Statement
Consider a sequence of real numbers defined by:
\begin{align*}
x_{1} & = c \\
x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \text{for all } n \geq 1.
\end{align*}
Show that if is a positive integer, then is an integer for all . (South Africa)